Part (REF ) follows along the lines of the proof of [1]}. Given a quasi-stable tropical curve \(\Gamma \)  over a monoid \(P\)  together with an admissible divisor \(D\)  on \(\mathbb {G}(\Gamma )\) , we can always find a quasi-stable logarithmic curve \(X\)  together with an admissible line bundle \(L\)  such that the dual tropical curve of \(X\)  is equal to \(\Gamma \)  and the multidegree of \(L\)  is equal to \(D\) . This shows that \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\)  is surjective.
In order to show that \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\)  is strict, we need to check that for every scheme \(S\)  together with a morphism \(M_S\rightarrow M_S^{\prime }\)  of logarithmic structures, any diagram
\(\begin{tikzcd}(S,M_S)[rr][d] &&\mathcal {J}_{g,n}^{log}[d,"\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}"]\\(S,M_S^{\prime })[rr][rru,dashed]&&\widetilde{\mathcal {J}}_{g,n}^{trop}\end{tikzcd}\) 
has a unique lift as pictured. In other words: We have a pair \((\Gamma ^{\prime },D^{\prime })\)  consisting of a quasi-stable tropical curve over \((S,M_S^{\prime })\)  and an admissible divisor \(D^{\prime }\)  on \(\Gamma ^{\prime }\)  and a pair \((X,M_X, L)\)  consisting of a quasi-stable logarithmic curve \((X,M_X)\)  over \((S,M_S)\)  and an admissible line bundle \(L\)  such that the family \(\Gamma _X\)  of dual tropical curves of \(X\)  over \(S\)  is the family \(\Gamma /(S,M_S)\)  of tropical curves that is induced from \(\Gamma ^{\prime }/(S,M_S)\) . We wish to show that there is a quasi-stable logarithmic curve \((X,M_X^{\prime })\)  over \((S,M_S^{\prime })\)  and an admissible line bundle \(L^{\prime }\)  on \(X\)  which induces both \((X,L)\)  over \((S,M_S)\)  and \((\Gamma ^{\prime },D^{\prime })\)  in \(\widetilde{\mathcal {J}}_{g,n}^{trop}(S,M_S^{\prime })\) . From the proof of [1]} we obtain such an \((X,M_X^{\prime })\)  by modifying the logarithmic structure on \((X,M_X)\)  and we set \(L^{\prime }=L\)  in order to find an \((X,M_X^{\prime },L^{\prime })\)  with the desired properties.
Since \(\widetilde{\mathcal {J}}_{g,n}^{trop}\)  is logarithmically étale over the base, we have that \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\)  is logarithmically smooth, since \(\mathcal {J}_{g,n}^{log}\)  is logarithmically smooth over \(k\) . Since \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\)  is also strict, the map \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\)  is smooth (in the classical non-logarithmic sense), since strict logarithmically smooth morphisms are automatically smooth.
Let us now prove part (REF ). It follows from [3]} that \( \mathcal {A}_{J_{g,n}^{log}}\)  is equal to the Artin fan associated, via the equivalence (REF ), to the (combinatorial) cone stack \(\Gamma _{\operatorname{Str}(\overline{\mathcal {J}}_{g,n})}:\operatorname{Str}(\overline{\mathcal {J}}_{g,n})\rightarrow \operatorname{\bf {RPC}}^f\)  of (REF ). On the other hand, since \(\widetilde{\mathcal {J}}^{trop}_{g,n}\)  is the Artin fan associated to the (combinatorial) cone stack \(\Gamma _{\operatorname{\mathcal {QD}iv}_{g,n}}:\operatorname{\mathcal {QD}iv}_{g,n}^{\operatorname{opp}}\rightarrow \operatorname{\bf {RPC}}^f\)  of (REF ) (by Proposition REF ), it follows from [3]} that \(\mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}\)  is the Artin fan associated to the (combinatorial) cone stack \(\Gamma _{\operatorname{\mathcal {QD}iv}_{g,n}^E}:(\operatorname{\mathcal {QD}iv}_{g,n}^E)^{\operatorname{opp}}\rightarrow \operatorname{\bf {RPC}}^f\)  of (REF ). Proposition REF , together with the modular description of \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\) , implies that the map \(\mathcal {A}(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}):\mathcal {A}_{J_{g,n}^{log}}\rightarrow \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}\)  is an isomorphism of Artin fans.
Let us prove part (REF ). The commutativity of the left square of (REF ) has been already observed in (REF ).
We now define a morphism \(\mathcal {A}(\widetilde{\Phi }^{trop}): \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}\rightarrow \mathcal {A}_{\widetilde{\mathcal {M}}^{trop}_{g,n}}\)  of Artin fans making commutative the central square of (REF ).
Consider the commutative diagram of combinatorial cone stacks
\({\left(\Gamma _{\operatorname{\mathcal {QD}iv}_{g,n}}: \operatorname{\mathcal {QD}iv}_{g,n}^{\operatorname{opp}} \longrightarrow \operatorname{\bf {RPC}}^f \right) [r] [d] & \left(\Gamma _{\operatorname{\mathcal {QD}iv}_{g,n}^E}: (\operatorname{\mathcal {QD}iv}_{g,n}^E)^{\operatorname{opp}} \longrightarrow \operatorname{\bf {RPC}}^f\right) [d] \\\left(\Gamma _{\operatorname{\mathcal {SG}}_{g,n}}: \operatorname{\mathcal {SG}}_{g,n}^{\operatorname{opp}} \longrightarrow \operatorname{\bf {RPC}}^f \right) [r] & \left(\Gamma _{\operatorname{\mathcal {SG}}_{g,n}^E}: (\operatorname{\mathcal {SG}}_{g,n}^E)^{\operatorname{opp}} \longrightarrow \operatorname{\bf {RPC}}^f\right)}\) 
where the two horizontal arrows are the natural factorizations of \(\Gamma _{\operatorname{\mathcal {QD}iv}_{g,n}}\)  (see (REF )) and \(\Gamma _{\operatorname{\mathcal {SG}}_{g,n}}\)  (see (REF )) through the quotient categories in Definition REF , the left vertical arrow is given in Theorem REF (REF ) and the right vertical arrow is the induced morphism.
Passing to the Artin fans associated to the above combinatorial cone stacks, we get the required commutative diagram
\({\widetilde{\mathcal {J}}^{trop}_{g,n} [rr]^{\operatorname{trop}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}} [d]_{\widetilde{\Phi }^{trop}} && \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}} [d]^{\mathcal {A}(\widetilde{\Phi }^{trop})} \\\widetilde{\mathcal {M}}^{trop}_{g,n} [rr]^{\operatorname{trop}_{\widetilde{\mathcal {M}}^{trop}_{g,n}}} && \mathcal {A}_{\widetilde{\mathcal {M}}^{trop}_{g,n}} \\}\) 
where we used the description of \(\widetilde{\Phi }^{trop}\)  in Proposition REF  and Theorem REF (REF ), the description of \(\operatorname{trop}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}\)  given in part (REF ) and the analogous description of \(\operatorname{trop}_{\widetilde{\mathcal {M}}^{trop}_{g,n}}\)  given in [3]}.
Finally, since the morphisms \(\mathcal {A}(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}})\)  and \(\mathcal {A}(\widetilde{\operatorname{trop}}_{\mathcal {M}_{g,n}^{log}})\)  are isomorphisms of Artin fans, there exists a unique morphism \(\mathcal {A}(\Phi ^{log}): \mathcal {A}_{J_{g,n}^{log}}\rightarrow \mathcal {A}_{\mathcal {M}_{g,n}^{log}}\)  making commutative the right square of (REF ).
Part (REF ): the first assertion follows from the modular description of \(\Upsilon _{J_{g,n}^{log}}\)  given in Proposition REF  and the definition of \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\) .
The second assertion follows from the first one together with Remarks REF  and REF .
